New constraints on exotic spin-dependent interactions with an ensemble-NV-diamond magnetometer

ABSTRACT Laboratory search of exotic interactions is crucial for exploring physics beyond the standard model. We report new experimental constraints on two exotic spin-dependent interactions at the micrometer scale based on ensembles of nitrogen-vacancy (NV) centers in diamond. A thin layer of NV electronic spin ensembles is synthesized as the solid-state spin quantum sensor, and a lead sphere is taken as the interacting nucleon source. Our result establishes new bounds for two types of exotic spin interactions at the micrometer scale. For an exotic parity-odd spin- and velocity-dependent interaction, improved bounds are set within the force range from 5 to 500 μm. The upper limit of the corresponding coupling constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$g_A^eg_V^N$\end{document} at 330 μm is more than 1000-fold more stringent than the previous constraint. For the P, T-violating scalar-pseudoscalar nucleon-electron interaction, improved constraints are established within the force range from 6 to 45 μm. The limit of the corresponding coupling constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$g_S^Ng_P^e$\end{document} is improved by more than one order of magnitude at 30 μm. This work demonstrates that a solid-state NV ensemble can be a powerful platform for probing exotic spin-dependent interactions.


I. NUMERICAL CALCULATION OF THE EFFECTIVE MAGNETIC FIELDS
In this section, we perform numerical calculation of the possible effective magnetic fields due to the exotic spin-dependent interactions. The effective magnetic fields between electron spin and nucleon are shown in Eq. (3) and (4) in the main text. By Integrating over the volume of both the lead sphere and NV layer, we derive the possible effective magnetic fields B AV and B SP sensed by the NV ensemble as follows, where ρ M = 6.8 × 10 30 m −3 is the nucleon density of the lead sphere. V S and V M are integral volume of NV layer and lead sphere, respectively. The radius of the lead sphere is R = 978(3) µm. The size of NV layer is 660 × 661 × 23 µm 3 . The minimal distance d 0 is 9. 3(5) µm.
The Monte Carlo method is utilized to numerically calculate the effective magnetic fields to avoid complex calculations due to high integral dimensions [1]. The algorithm of Monte Carlo integral is performed as follows: (1) N M C = 2 20 random pairs of points inside both the volumes of the lead sphere and the NV ensemble are generated.
(2) The effective magnetic field B i ef f, AV (B i ef f, SP ) between a randomly generated pair of points is calculated with a given force range.
B i ef f, SP = g N S g e P 4πm e γ e ( 1 λr where θ = arccos(1/ √ 3) is the angle between the direction of the velocity v and the NV axis.
(3) All the contributions to the effective magnetic fields are summed and normalized to give the average magnetic fields generated by the lead sphere and sensed by the NV ensemble: where N nucleon is the total number of nucleons in lead sphere.
The magnetic field B AV and B SP can be decomposed into orthogonal components of Fourier series as shown in Eq. (5) and (6) in the main text, the coefficients can be derived as a (n) where T = 1/f M is the period of the effective magnetic field. We take g e A g N V = 1 × 10 −20 , and λ = 10 −4 m as an example, the coefficients of B AV at first three harmonic frequencies are listed as follows AV (pT) 9.62 0.02 0.00 The amplitude of the first harmonic coefficient b (1) AV is much larger than higher order harmonic coefficients b AV and b AV . Values of a (n) AV are zero. Similarly, the calculated coefficients of B SP at the first three harmonic frequencies are   SP is much larger than higher order harmonic coefficients a

II. THE PERFORMANCE OF THE ENSEMBLE-NV-DIAMOND MAGNETOMETER
The ensemble-NV-diamond magnetometer used in our experiment was based on the continuous-wave (CW) method, wherein laser and microwave fields are continuously applied to NV centers. The NV centers of |m s = 0 state can be transmitted to |m s = +1 by a resonance microwave with an angular frequency ω e = D + γ e B 0 , where D = 2π × 2.87 GHz is the ground-state zero splitting, γ e = 2π × 28 GHz/T is the gyromagnetic ratio of the electron spin, and B 0 is the bias magnetic field along the symmetry axis of NV centers generated by a solenoid coil. When the external magnetic field varies, the population on |m s = +1 states decreases, resulting in changes in fluorescence which can be detected.
In order to avoid flicker noise, the frequency modulation technique was used in our experiment. The frequency of microwave from the synthesizer was modulated by a lock-in amplifier (LIA1 in Fig. 2 of the main text) with modulation frequency FM being 87.975 kHz. The signal of PD, which detected the fluorescence from NV centers, was demodulated with the same frequency. For further noise cancellation, the signal of a reference PD used to monitor the power fluctuation of the laser was also demodulated by LIA1 with a frequency of 87.975 kHz. The laser intensity noise was canceled by scaling and subtracting the demodulated reference signal from the demodulated fluorescence signal, with a cancellation coefficient of about 2 [3].
The calibrated constant of the magnetic field to the output of the magnetometer was determined by the max slope of the CW spectrum, which was 0.816 ± 0.009 V/MHz as shown in Supplementary Fig. S1(a), and corresponds to a calibrated constant of (2.29 ± 0.03) × 10 4 V/T with the gyromagnetic ratio γ e = 2π × 28 GHz/T. The sensitivity of 1.4 nT/Hz 1/2 from 0.4 to 2 kHz was achieved, as shown in Supplementary  Fig. S1(b).
Our experiment measured the variation in amplitude of the external magnetic field at f M = 1.953 kHz under a static bias magnetic field B 0 . As shown in Supplementary  Fig. S1(b), the frequency component with frequency f M in B 0 is clean. Furthermore, our measurement was in phase with the vibration of the lead sphere. The variation of B 0 was asynchronous with the vibration. During 291.9-hour experimental measurement, the frequency component at 1.953 kHz in B 0 did not occur due to uncorrelated random phase. Besides, the possible minor noise of B 0 at f M was included in the measurement result together with the target effective magnetic fields. Our final zero result also showed no effect of the variation of B 0 at f M . In conclusion, the possible minor noise of B 0 at 1.953 kHz is negligible in our experiment.

III. CALIBRATION OF THE PHASE DELAY
The phase delay φ between the output signal of the ensemble-NV-diamond magnetometer and d(t) can be calibrated by a given signal with a method similar to that described in Ref. [2]. The calibration procedure was carried out before the experiments with the lead sphere. A thin copper wire carrying a DC current was stuck to the front section of the piezoelectric bender. The magnetic field generated by the current-carrying copper wire was modulated by the vibration of the piezoelectric bender and thus in phase with d(t). The output signal of the magnetometer and the feedback of the piezoelectric bender, which was used to monitor d(t), were demodulated by a lock-in amplifier (LIA2 in Fig. 2 of the main text) with the same frequency f M = 1.953 kHz. The demodulation signal from LIA2 provided the amplitude of the magneic field as (18 ± 2) nT. The phase delay between the demodulation signal of the output signal of magnetometer and feedback of piezoelectric bender was φ 1 = −32(9) • .
We also use a commercial laser vibrometer (Sunny Optical, LV-S01) to measure d(t). The phase delay between d(t) and feedback of piezoelectric bender was calibrated to be φ 2 = −86(1) • . The phase delay φ between the output signal of the magnetometer and d(t) can be obtained to be φ = φ 1 − φ 2 = 54(9) • . For experimental operation, considering d(t) was monitored by the feedback of piezoelectric bender, the lock-in amplifier (LIA2) was set with a phase difference ∆φ = φ 1 = −32 • between the demodulation phase of the output signal of magnetometer and feedback of piezoelectric bender.

IV. LIST OF EXPERIMENT INSTRUMENTS
The schematic of the experimental setup is shown in Fig. 2 in the main text. Table I shows the manufacturers and models of devices used in our experiment.

Diamagnetism of the lead sphere
The diamagnetism of the vibrating lead sphere leads to the modulation of a static magnetic field sensed by the NV ensemble. Both DC component and AC component of the effect of diamagnetism are unobservable in our experiment according to following detailed analyses.
We first calculate the static magnetic field due to diamagnetism among the sensing area of the NV ensemble. With a bias magnetic field of B 0 = 20 Gauss, the diamagnetism of the lead sphere causes a magnetic field B diam on each NV center in the layer, where χ = −16 × 10 −6 , is magnetic susceptibility of the lead sphere [4], V M is the integral volume of lead sphere. Since the large zero filed splitting of NV centers, the magnetic field perpendicular to NV axis can be ignored. The magnetic field parallel to NV axis due to diamagnetism is denoted as B diam, . The distribution of B diam, in NV ensemble is shown in Supplementary Fig.  S2. The DC component of the effect of the diamagnetism leads to an inhomogeneous static magnetic field among the NV ensemble layer. The maximum variation of the magnetic field is 14 nT. Since the measurement of the effective magnetic field shown in Fig. 3 in the main text is obtained after demodulation at the specific frequency f M = 1.953 kHz, the DC component of the magnetic field is filtered out. The static magnetic field due to diamagnetism may induce an NV ensemble CW spectrum linewidth broadening of less than 0.4 kHz across the sensing area, which is much smaller than the megahertz linewidth in our experiment. Therefore, the DC component of the effect of diamagnetism is negligible in our experiment.
Then we calculate the AC component of the magnetic field caused by the diamagnetism of the vibrating lead sphere. The averaged magnetic field sensed by the NV ensemble due to diamagnetism is calculated to be in the range from 0.738 pT to 0.740 pT during the vibration of the lead sphere, corresponding to an AC magnetic field with the amplitude being 0.001 pT, which is much less than the standard error of the measured field under current statistics. We also take possible misalignment of the lead sphere and diamond into consideration. The maximum misalignment is estimated to be 10 µm. The linewidth broadening caused by the static magnetic field due to diamagnetism is less than 0.4 kHz and is negligible in our experiment. The magnetic field sensed by the NV ensemble due to diamagnetism is calculated to be in the range from 200.4 pT to 201.4 pT during the vibration of the lead sphere, corresponding to an AC magnetic field with the amplitude being 0.5 pT, which is less than the standard error of the measured field under current statistics.
In conclusion, both DC component and AC component of the effect of diamagnetism are unobservable in our experiment. The magnetic field due to diamagnetism is in phase with d(t) and could only appear in the in-phase component rather than the quadrature component of our measurement. This may affect the result of B SP rather than that of B AV . The correction to g N S g e P is (0.0±2.9)× 10 −21 at λ = 30 µm. Taking uncertainty in phase delay φ into account, the correction to g e A g N V is (0.0±0.3)×10 −25 at λ = 330 µm.

Uncertainty in d 0
The distance between the bottom of the lead sphere and the diamond is adjusted by a vertically installed piezo motor (Physik Instrumente, Q-545). The lead sphere first slowly approaches the diamond surface with a tiny vibration amplitude, and the position of slight contact can be detected when the feedback of piezoelectric bender suddenly varies [5]. The lead sphere was then lifted by 10 µm, according to the integrated position sensor of the piezo motor. The minimal distance between the bottom of the lead sphere and the diamond d 0 is 9.3(5) µm since the vibration amplitude is A = 718(7) nm. The uncertainty is due to long time drift of our system.
To estimate the corrections to g e A g N V and g N S g e P due to the uncertainty in d 0 , 10 5 samples for d 0 was randomly taken, which satisfied a Gaussian distribution P d0 (d 0,i ) = ]. µ d0 = 9.3 µm and σ d0 = 0.5 µm are the mean value and uncertainty of measured d 0 . Then the mean values and the standard deviations of g e A g N V and g N S g e P can be obtained. The correction to g e A g N V is (0.0 ± 0.2) × 10 −25 at λ = 330 µm. The correction to g N S g e P is (0.0 ± 0.4) × 10 −21 at λ = 30 µm.

Uncertainty in R
The radius of the lead sphere is measured to be R = 978(3) µm. The correction to g e A g N V is obtained in the same procedure as that of uncertainty in d 0 . The correction to g e A g N V is (0.0±0.2) × 10 −25 at λ = 330 µm. The correction to g N S g e P is (0.0±0.3) × 10 −21 at λ = 30 µm.

Uncertainty in θ
The angle between the effective magnetic field and the NV axis is 54.7 ± 1.3 • , containing crystallographic orientation of the diamond and the angle between the surface of the diamond and the piezoelectric bender. The uncertainty of θ mainly comes from the tilt of the piezoelectric bender, which can be obtained through our optical system. The correction to g e A g N V is from −2.8×10 −25 to 2.9×10 −25 at λ = 330 µm. The correction to g N S g e P is (0.0±0.4) × 10 −21 at λ = 30 µm.

Uncertainty in h
The thickness of the NV layer is estimated by measuring the thickness of the diamond before and after the growth of the NV layer. The original thickness of the diamond is 551(1) µm. After the growth of the NV layer, the thickness of the diamond is 574(1) µm. The thickness of the NV layer is measured to be h = 23(1) µm. The correction to g e A g N V is (0.0±0.2) × 10 −25 at λ = 330 µm. The correction to g N S g e P is from −0.4×10 −21 to 0.3×10 −21 at λ = 30 µm.

Uncertainty in A
The vibration amplitude is measured to be A = 718(7) nm, using a commercial laser vibrometer (Sunny Optical, LV-S01). The correction to g e A g N V is from −1.0×10 −25 to 0.8×10 −25 at λ = 330 µm. The correction to g N S g e P is −0.4 × 10 −21 to 0.3 × 10 −21 at λ = 30 µm. Deviation in x-y plane.
The deviation in x-y plane is measured to be (0 ± 10) µm according to the CCD images as shown in Fig.  S3. The correction to g e A g N V is (0.0±0.2) × 10 −25 at λ = 330 µm. The correction to g N S g e P is −0.4 × 10 −21 to 0.3 × 10 −21 at λ = 30 µm.
The calibrated constant of the magnetic field to the output of magnetometer is measured to be (2.29±0.03)× 10 4 V/T. The correction to g e A g N V is (0.0±1.2) × 10 −25 at λ = 330 µm. The correction to g N S g e P is (0.0±0.3)×10 −21 at λ = 30 µm.
With the fiducial probability of 95%, the upper bound |g e A g N V | ≤ 2.5 × 10 −22 for the force range λ = 330 µm and the upper bound |g N S g e P | ≤ 2.5 × 10 −20 for the force range λ = 30 µm were obtained, taking both statistical and systematic errors into account.